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Considering such multiplication operators $M_B$, we present a representation of those operators commuting with both $M_B$ and $M_B^*$. It is shown that for \"most\" thin Blaschke products $B$, $M_B$ is irreducible, i.e. $M_B$ has no nontrivial reducing subspace; and such a thin Blaschke product $B$ is constructed.\n  As an application of the methods, it is proved that for \"most\" finite Blaschke products $\\p","authors_text":"Hansong Huang, Kunyu Guo","cross_cats":["math.OA"],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.FA","submitted_at":"2013-06-30T03:45:52Z","title":"Geometric constructions of thin Blaschke products and reducing subspace problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0174","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1fc3a76731d0793f8318c2a03042e25da880296a4845a27886c07367dbbf0266","target":"record","created_at":"2026-05-18T00:44:29Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"37997997859aff8f6f3830b2b0a911505df87f3b0d38ef7e51cc61826b25a244","cross_cats_sorted":["math.OA"],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.FA","submitted_at":"2013-06-30T03:45:52Z","title_canon_sha256":"d599675d2dab9b37a5594a0d800c4d6683ee0ae65ec0c3afd20f1dab34550112"},"schema_version":"1.0","source":{"id":"1307.0174","kind":"arxiv","version":1}},"canonical_sha256":"1afce79161c3bdbae38441999e18da2287ac602ea279bd9487dfcff1e78734d6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1afce79161c3bdbae38441999e18da2287ac602ea279bd9487dfcff1e78734d6","first_computed_at":"2026-05-18T00:44:29.672744Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:29.672744Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NAwjnV7EFRhzwhC/dyUno67uf+K4goVqYw14gYHd9tkujfXZ8fH0K85OdS3SZkxlvIxCxEzbgzhtafDJieGcCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:29.673272Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.0174","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1fc3a76731d0793f8318c2a03042e25da880296a4845a27886c07367dbbf0266","sha256:ef3db2ee5ff12962a16638f7aa8c92ff8d430a81a30c59aae60a86327debbbe2"],"state_sha256":"e9447d62c7f69b6a9c005409fa005c9392873f8fdc88af43d4e9b432c1bd2dad"}